The utility of DGSPT was demonstrated in Section  in the context of three non-trivial applications. In Section REF  we used DGSPT in order to provide a geometric analysis of the Chialvo map-based model for neuronal bursting in a fast-slow parameter regime, thereby validating and extending previous work in [1]}, [2]}, [3]}. The DGSPT formalism provided a systematic approach to the identification of distinct singular structure corresponding to four types of neuronal dynamics: excitability, relaxation, regular (non-chaotic) bursting and chaotic bursting, recall Figure REF . In Section REF  we used DGSPT to prove results on the relationship between the geometry and dynamics of fast-slow ODEs in the general (non-standard) form (REF ) and its corresponding Euler discretization, see Theorem REF . These results extend pre-existing results on the discretization of fast-slow ODEs in standard form (REF ) in e.g. [4]}, [5]}. Finally in Section REF , we showed that fast-slow ODE systems with a single slow variable (which often arise in standard form (REF ) after allowing a parameter to evolve slowly in time) give rise to fast-slow Poincaré maps with a normally hyperbolic critical manifold if the layer problem has a hyperbolic limit cycle, see Lemma REF . This critical manifold, which lies within the intersection of a limit cycle manifold \(\mathcal {M}\)  of the (continuous-time) layer problem and a transverse section \(\Delta \) , perturbs to a nearby slow manifold for the Poincaré map for \(0 < \varepsilon \ll 1\) . The geometry and dynamics of the Poincaré map are described in detail in Theorem REF , which follows after a direct application of the invariant manifold theorems in Section . Properties of the Poincaré map were then applied in order to prove Theorem REF , which describes the persistence of the limit cycle manifold \(\mathcal {M}\)  as a nearby locally invariant manifold \(\mathcal {M}_\varepsilon \)  in the ODE and, additionally, provides a criteria for the existence and hyperbolicity of limit cycles within \(\mathcal {M}_\varepsilon \) .
